Simulation of a chain of collapsible contracting lymphangions with progressive valve closure

Computational fluid dynamic modeling of the lymphatic system: a review of existing models and future directions

Historically, research into the lymphatic system has been overlooked due to both a lack of knowledge and limited recognition of its importance. In the last decade however, lymphatic research has gained substantial momentum and has included the development of a variety of computational models to aid understanding of this complex system. This article reviews existing computational fluid dynamic models of the lymphatics covering each structural component including the initial lymphatics, pre-collecting and collecting vessels, and lymph nodes. This is followed by a summary of limitations and gaps in existing computational models and reasons that development in this field has been hindered to date. Over the next decade, efforts to further characterize lymphatic anatomy and physiology are anticipated to provide key data to further inform and validate lymphatic fluid dynamic models. Development of more comprehensive multiscale- and multi-physics computational models has the potential to significantly enhance the understanding of lymphatic function in both health and disease.

A 1D model characterizing the role of spatiotemporal contraction distributions on lymph transport

Lymphedema is a condition in which lymph transport is compromised. The factors that govern the timing of lymphatic contractions are largely unknown; however, these factors likely play a central role in lymphatic health. Computational models have proven useful in quantifying changes in lymph transport; nevertheless, there is still much unknown regarding the regulation of contractions. The purpose of this paper is to utilize computational modeling to examine the role of pacemaking activity in lymph transport. A 1D fluid-solid modeling framework was utilized to describe the interaction between the contracting vessel and the lymph flow. The distribution of contractions along a three-lymphangion chain in time and space was determined by specifying the pacemaking sites and parameters obtained from experimentation. The model effectively replicates the contractility patterns in experiments. Quantitatively, the flow rates were measured at 5.44 and 2.29 [Formula: see text], and the EF values were 78% and less than 33% in the WT and KO models, respectively, which are consistent with the literature. Applying pacemaking parameters in this modeling framework effectively captures lymphatic contractile wave propagations and their relation to lymph transport. It can serve as a motivation for conducting novel studies to evaluate lymphatic pumping function during the development of lymphedema.

Potentiating glymphatic drainage minimizes post-traumatic cerebral oedema

Cerebral oedema is associated with morbidity and mortality after traumatic brain injury (TBI)1. Noradrenaline levels are increased after TBI2-4, and the amplitude of the increase in noradrenaline predicts both the extent of injury5 and the likelihood of mortality6. Glymphatic impairment is both a feature of and a contributor to brain injury7,8, but its relationship with the injury-associated surge in noradrenaline is unclear. Here we report that acute post-traumatic oedema results from a suppression of glymphatic and lymphatic fluid flow that occurs in response to excessive systemic release of noradrenaline. This post-TBI adrenergic storm was associated with reduced contractility of cervical lymphatic vessels, consistent with diminished return of glymphatic and lymphatic fluid to the systemic circulation. Accordingly, pan-adrenergic receptor inhibition normalized central venous pressure and partly restored glymphatic and cervical lymphatic flow in a mouse model of TBI, and these actions led to substantially reduced brain oedema and improved functional outcomes. Furthermore, post-traumatic inhibition of adrenergic signalling boosted lymphatic export of cellular debris from the traumatic lesion, substantially reducing secondary inflammation and accumulation of phosphorylated tau. These observations suggest that targeting the noradrenergic control of central glymphatic flow may offer a therapeutic approach for treating acute TBI.

An Enhanced 3D Model of Intravascular Lymphatic Valves to Assess Leaflet Apposition and Transvalvular Differences in Wall Distensibility

Lymphatic valves operate in a fluid-dynamically viscous environment that has little in common with that of cardiac valves, and accordingly have a different, axially lengthened, shape. A previously developed 3D fluid/structure interaction model of a lymphatic valve was extended to allow the simulation of stages of valve closure after the leaflets come together. This required that the numerical leaflet be prevented from passing into space occupied by the similar other leaflet. The resulting large deflections of the leaflet and lesser deflections of the rest of the valve were mapped as functions of the transvalvular pressure. In a second new development, the model was reconstructed to allow the vessel wall to have different material properties on either side of where the valve leaflet inserts into the wall. As part of this, a new pre-processing scheme was devised which allows easier construction of models with modified valve dimensions, and techniques for successfully interfacing the CAD software to the FE software are described. A two-fold change in wall properties either side of the leaflet made relatively little difference to valve operation apart from affecting the degree of sinus distension during valve closure. However, the numerically permitted strains were modest (<14%), and did not allow examination of the large-scale highly nonlinear elastic properties exhibited by real lymphatic vessels. A small series of murine popliteal, mesenteric, and inguinal-axillary lymphatic vessel segments containing a valve were experimentally investigated ex vivo. The pressure–diameter curves measured just upstream and just downstream of the valve were parameterised by computing the difference in tubular distensibility at three values of transmural pressure. In the popliteal and mesenteric segments, it was found that the distensibility was usually greater just downstream, i.e., in the sinus region, than upstream, at low and intermediate transmural pressure. However, there was wide variation in the extent of difference, and possible reasons for this are discussed.

Lymphatic Function Decreases over Time in the Arms of Breast Cancer Patients following Treatment

In patients with breast cancer-related lymphedema, distinct lymphatic patterns and changed lymphatic contractile function have been described, but it is unknown how these characteristics change over time and to what extent they appear before clinical edema is detectable. Recently, we described the lymphatic morphology and function in a cohort of breast cancer patients shortly after radiation therapy (RT). In the current study, we investigate lymphatic function and morphology in the same cohort after 1 year of follow-up.

Methods

The study population consisted of 28 breast cancer patients investigated 12 months after adjuvant locoregional RT. Lymphatic contraction frequency (CF), propulsion velocity, and the morphology of lymphatic vessels in the upper extremities were described in vivo using near-infrared fluorescence imaging. Lymphatic stress test was performed using hyperthermia.

Results

At 1 year after RT, (n = 28) 46% of the patients presented with lymphatic morphological abnormalities with a degree of dermal backflow and 21% had developed clinical breast cancer-related lymphedema. In the ipsilateral arm, CF was 23% lower than in the contralateral arm (P = 0.04). Since primary examination, CF in the ipsilateral arm decreased by 40% (P = 0.03), whereas no change was observed in the contralateral arm. During hyperthermia, the ipsilateral arms with lymphatic complications were not able to increase CF as the remaining subgroups.

Conclusions

Lymphatic function in the ipsilateral arm deteriorated over time after adjuvant breast cancer therapy. Furthermore, the presence of abnormal torturous lymphatic vessels in asymptomatic arms appeared to be associated with weak lymphatic reserve pumping capacity.

A multiscale sliding filament model of lymphatic muscle pumping

The lymphatics maintain fluid balance by returning interstitial fluid to veins via contraction/compression of vessel segments with check valves. Disruption of lymphatic pumping can result in a condition called lymphedema with interstitial fluid accumulation. Lymphedema treatments are often ineffective, which is partially attributable to insufficient understanding of specialized lymphatic muscle lining the vessels. This muscle exhibits cardiac-like phasic contractions and smooth muscle-like tonic contractions to generate and regulate flow. To understand the relationship between this sub-cellular contractile machinery and organ-level pumping, we have developed a multiscale computational model of phasic and tonic contractions in lymphatic muscle and coupled it to a lymphangion pumping model. Our model uses the sliding filament model (Huxley in Prog Biophys Biophys Chem 7:255-318, 1957) and its adaptation for smooth muscle (Mijailovich in Biophys J 79(5):2667-2681, 2000). Multiple structural arrangements of contractile components and viscoelastic elements were trialed but only one provided physiologic results. We then coupled this model with our previous lumped parameter model of the lymphangion to relate results to experiments. We show that the model produces similar pressure, diameter, and flow tracings to experiments on rat mesenteric lymphatics. This model provides the first estimates of lymphatic muscle contraction energetics and the ability to assess the potential effects of sub-cellular level phenomena such as calcium oscillations on lymphangion outflow. The maximum efficiency value predicted (40%) is at the upper end of estimates for other muscle types. Spontaneous calcium oscillations during diastole were found to increase outflow up to approximately 50% in the range of frequencies and amplitudes tested.

Mathematical Modeling of Lymph Node Drainage Function by Neural Network

The lymph node (LN) represents a key structural component of the lymphatic system network responsible for the fluid balance in tissues and the immune system functioning. Playing an important role in providing the immune defense of the host organism, LNs can also contribute to the progression of pathological processes, e.g., the spreading of cancer cells. To gain a deeper understanding of the transport function of LNs, experimental approaches are used. Mathematical modeling of the fluid transport through the LN represents a complementary tool for studying the LN functioning under broadly varying physiological conditions. We developed an artificial neural network (NN) model to describe the lymph node drainage function. The NN model predicts the flow characteristics through the LN, including the exchange with the blood vascular systems in relation to the boundary and lymphodynamic conditions, such as the afferent lymph flow, Darcy’s law constants and Starling’s equation parameters. The model is formulated as a feedforward NN with one hidden layer. The NN complements the computational physics-based model of a stationary fluid flow through the LN and the fluid transport across the blood vessel system of the LN. The physical model is specified as a system of boundary integral equations (IEs) equivalent to the original partial differential equations (PDEs; Darcy’s Law and Starling’s equation) formulations. The IE model has been used to generate the training dataset for identifying the NN model architecture and parameters. The computation of the output LN drainage function characteristics (the fluid flow parameters and the exchange with blood) with the trained NN model required about 1000-fold less central processing unit (CPU) time than computationally tracing the flow characteristics of interest with the physics-based IE model. The use of the presented computational models will allow for a more realistic description and prediction of the immune cell circulation, cytokine distribution and drug pharmacokinetics in humans under various health and disease states as well as assisting in the development of artificial LN-on-a-chip technologies.

Lymphatic Function in the Arms of Breast Cancer Patients-A Prospective Cohort Study

Background:

Lymphedema is a highly feared complication of breast cancer treatment, but the underlying complex mechanisms are still unknown. Thus, we investigated the lymphatic morphology and contractility in the lymphatic vessels of arms of high-risk breast cancer patients treated for node-positive early breast cancer.

Methods:

In this prospective cohort study 32 women treated for unilateral node-positive breast cancer were enrolled and studied 36 ± 23 days after loco-regional radiotherapy. Near-infrared fluorescence imaging was used to assess morphology and function of the superficial lymphatic vessels. Strain-gauge plethysmography was performed to evaluate the capillary filtration of fluid.

Both arms were investigated, with the non-treated arm acting as control. The patients were questioned about the presence of lymphedema yearly and finally 574 ± 118 days after ended radiotherapy.

Results:

Morphologically, 25% of the treated arms expressed lymphatic vessel abnormalities compared to the control arms (p = 0.0048). No difference in functional parameters (maximal pumping pressure, p = 0.20; contraction frequency, p = 0.63; contraction velocity, p = 0.55) was found between the treated and control arms. Patients who later developed lymphedema had a difference in velocity compared to those who did not develop lymphedema (p = 0.02). The capillary filtration rate was similar between the two arms (p = 0.18).

Conclusions:

Peripheral lymphatic vessels were morphologically changed in the ipsilateral arm in 25% of the patients and patients who later developed lymphedema showed an early increase in velocity. Other functional parameters and capillary filtration were unchanged in this early phase. These discrete changes might be early indicators of later development of lymphedema.

Pump efficacy in a two-dimensional, fluid-structure interaction model of a chain of contracting lymphangions

The transport of lymph through the lymphatic vasculature is the mechanism for returning excess interstitial fluid to the circulatory system, and it is essential for fluid homeostasis. Collecting lymphatic vessels comprise a significant portion of the lymphatic vasculature and are divided by valves into contractile segments known as lymphangions. Despite its importance, lymphatic transport in collecting vessels is not well understood. We present a computational model to study lymph flow through chains of valved, contracting lymphangions. We used the Navier-Stokes equations to model the fluid flow and the immersed boundary method to handle the two-way, fluid-structure interaction in 2D, non-axisymmetric simulations. We used our model to evaluate the effects of chain length, contraction style, and adverse axial pressure difference (AAPD) on cycle-mean flow rates (CMFRs). In the model, longer lymphangion chains generally yield larger CMFRs, and they fail to generate positive CMFRs at higher AAPDs than shorter chains. Simultaneously contracting pumps generate the largest CMFRs at nearly every AAPD and for every chain length. Due to the contraction timing and valve dynamics, non-simultaneous pumps generate lower CMFRs than the simultaneous pumps; the discrepancy diminishes as the AAPD increases. Valve dynamics vary with the contraction style and exhibit hysteretic opening and closing behaviors. Our model provides insight into how contraction propagation affects flow rates and transport through a lymphangion chain.

Fluid pumping of peristaltic vessel fitted with elastic valves

Using numerical simulations, we probe the fluid flow in an axisymmetric peristaltic vessel fitted with elastic bi-leaflet valves. In this biomimetic system that mimics the flow generated in lymphatic vessels, we investigate the effects of the valve and vessel properties on pumping performance of the valved peristaltic vessel. The results indicate that valves significantly increase pumping by reducing backflow. The presence of valves, however, increases the viscous resistance therefore requiring greater work compared to valveless vessels. The benefit of the valves is the most significant when the fluid is pumped against an adverse pressure gradient and for low vessel contraction wave speeds. We identify the optimum vessel and valve parameters leading to the maximum pumping efficiency. We show that the optimum valve elasticity maximizes the pumping flow rate by allowing the valve to block more effectively the backflow while maintaining low resistance during the forward flow. We also examine the pumping in vessels where the vessel contraction amplitude is a function of the adverse pressure gradient as found in lymphatic vessels. We find that in this case the flow is limited by the work generated by the contracting vessel, suggesting that the pumping in lymphatic vessels is constrained by the performance of lymphatic muscle. Given the regional heterogeneity of valve morphology observed throughout the lymphatic vasculature, these results provide insight into how these variations might facilitate efficient lymphatic transport in the vessel's local physiologic context.

Microfluidic valvular chips and a numerical lymphatic vessel model for the study of lymph transport characteristics

Lymph transport inside lymphatic vessels is highly complex and not yet fully understood. So far, a consensus has not been reached among existing analytical models on how spatiotemporal coordination of contracting adjacent lymphangions affects lymph transport. To understand complex lymph transport, we created a novel microfluidic valvular chip with flexible bicuspid valves and segmental pneumatic pumps based on a microfluidic device with an inside 3D structure made of hydrogels. Inside the chip, water moved unidirectionally when the microfluidic channel was locally compressed, with its velocity profile closely resembling the waveform of lymph observed in vivo. Furthermore, for a systematic and mechanistic study, we constructed a numerical model based on fluid-structure interaction and validated the model via demonstration of similarities in water transport characteristics between the model and the chip. Using this model, we examined various mechanical and time-dependent parameters, such as period, phase delay, sequence, and strength of contractions, valve compliance, fluid viscosity, and pressure differences, for their effects on water transport. Although our model is simplified, it enabled a parametric study that helped clarify the mechano-temporal correlations between compressions of adjacent chambers via transmissions of hydrodynamic forces, which regulate complex lymph transport. Moreover, our chip demonstrated technical advances that enable unidirectional discrete movement of fluid in the picoliter range by phenumatic pumping. The velocity profile is also similar to the pulse waveform of arteries under pathological conditions such as increased aortic stiffness, allowing our chip to be used for in vitro mechanobiology studies of endothelial cells.

Functional lymphatic reserve capacity is depressed in patients with a Fontan circulation

Background

Lymphatic abnormalities play a role in effusions in individuals with a Fontan circulation. Recent results using near‐infrared fluorescence imaging disclosed an increased contraction frequency of lymphatic vessels in Fontan patients compared to healthy controls. It is proposed that the elevated lymphatic pumping seen in the Fontan patients is necessary to maintain habitual interstitial fluid balance. Hyperthermia has previously been used as a tool for lymphatic stress test. By increasing fluid filtration in the capillary bed, the lymphatic workload and contraction frequency are increased accordingly. Using near‐infrared fluorescence imaging, the lymphatic functional reserve capacity in Fontan patients were explored with a lymphatic stress test.

Methods

Fontan patients (n = 33) were compared to a group of 15 healthy individuals of equal age, weight, and gender. The function of the superficial lymphatic vessels in the lower leg during rest and after inducing hyperthermia was investigated, using near‐infrared fluorescence imaging.

Results

Baseline values in the Fontan patients showed a 57% higher contraction frequency compared to the healthy controls (0.4 ± 0.3 min⁻¹ vs. 0.3 ± 0.2 min⁻¹, p = 0.0445). After inducing stress on the lymphatic vessels with hyperthermia the ability to increase contraction frequency was decreased in the Fontan patients compared to the controls (0.6 ± 0.5 min⁻¹ vs. 1.2 ± 0.8 min⁻¹, p = 0.0102).

Conclusions

Fontan patients had a higher lymphatic contraction frequency during normal circumstances. In the Fontan patients, the hyperthermia response is dampened indicating that the functional lymphatic reserve capacity is depressed. This diminished reserve capacity could be part of the explanation as to why some Fontan patients develop late‐onset lymphatic complications.

Characterization of rat tail lymphatic contractility and biomechanics: incorporating nitric oxide-mediated vasoregulation

The lymphatic system transports lymph from the interstitial space back to the great veins via a series of orchestrated contractions of chains of lymphangions. Biomechanical models of lymph transport, validated with ex vivo or in vivo experimental results, have proved useful in revealing novel insight into lymphatic pumping; however, a need remains to characterize the contributions of vasoregulatory compounds in these modelling tools. Nitric oxide (NO) is a key mediator of lymphatic pumping. We quantified the active contractile and passive biaxial biomechanical response of rat tail collecting lymphatics and changes in the contractile response to the exogenous NO administration and integrated these findings into a biomechanical model. The passive mechanical response was characterized with a three-fibre family model. Nonlinear regression and non-parametric bootstrapping were used to identify best-fit material parameters to passive cylindrical biaxial mechanical data, assessing uniqueness and parameter confidence intervals; this model yielded a good fit (R² = 0.90). Exogenous delivery of NO via sodium nitroprusside (SNP) elicited a dose-dependent suppression of contractions; the amplitude of contractions decreased by 30% and the contraction frequency decreased by 70%. Contractile function was characterized with a modified Rachev-Hayashi model, introducing a parameter that is related to SNP concentration; the model provided a good fit (R² = 0.89) to changes in contractile responses to varying concentrations of SNP. These results demonstrated the significant role of NO in lymphatic pumping and provide a predictive biomechanical model to integrate the combined effect of mechanical loading and NO on lymphatic contractility and mechanical response.

Mathematical Modelling of the Structure and Function of the Lymphatic System

This paper presents current knowledge about the structure and function of the lymphatic system. Mathematical models of lymph flow in the single lymphangion, the series of lymphangions, the lymph nodes, and the whole lymphatic system are considered. The main results and further perspectives are discussed.

The Role of Lymphatic Vascular Function in Metabolic Disorders

In addition to its roles in the maintenance of interstitial fluid homeostasis and immunosurveillance, the lymphatic system has a critical role in regulating transport of dietary lipids to the blood circulation. Recent work within the past two decades has identified an important relationship between lymphatic dysfunction and patients with metabolic disorders, such as obesity and type 2 diabetes, in part characterized by abnormal lipid metabolism and transport. Utilization of several genetic mouse models, as well as non-genetic models of diet-induced obesity and metabolic syndrome, has demonstrated that abnormal lymphangiogenesis and poor collecting vessel function, characterized by impaired contractile ability and perturbed barrier integrity, underlie lymphatic dysfunction relating to obesity, diabetes, and metabolic syndrome. Despite the progress made by these models, the contribution of the lymphatic system to metabolic disorders remains understudied and new insights into molecular signaling mechanisms involved are continuously developing. Here, we review the current knowledge related to molecular mechanisms resulting in impaired lymphatic function within the context of obesity and diabetes. We discuss the role of inflammation, transcription factor signaling, vascular endothelial growth factor-mediated signaling, and nitric oxide signaling contributing to impaired lymphangiogenesis and perturbed lymphatic endothelial cell barrier integrity, valve function, and contractile ability in collecting vessels as well as their viability as therapeutic targets to correct lymphatic dysfunction and improve metabolic syndromes.

Modelling secondary lymphatic valves with a flexible vessel wall: how geometry and material properties combine to provide function

A three-dimensional finite-element fluid/structure interaction model of an intravascular lymphatic valve was constructed, and its properties were investigated under both favourable and adverse pressure differences, simulating valve opening and valve closure, respectively. The shear modulus of the neo-Hookean material of both vascular wall and valve leaflet was varied, as was the degree of valve opening at rest. Also investigated was how the valve characteristics were affected by prior application of pressure inflating the whole valve. The characteristics were parameterised by the volume flow rate through the valve, the hydraulic resistance to flow, and the maximum sinus radius and inter-leaflet-tip gap on the plane of symmetry bisecting the leaflet, all as functions of the applied pressure difference. Maximum sinus radius on the leaflet-bisection plane increased with increasing pressure applied to either end of the valve segment, but also reflected the non-circular deformation of the sinus cross section caused by the leaflet, such that it passed through a minimum at small favourable pressure differences. When the wall was stiff, the inter-leaflet gap increased sigmoidally during valve opening; when it was as flexible as the leaflet, the gap increased more linearly. Less pressure difference was required both to open and to close the valve when either the wall or the leaflet material was more flexible. The degree of bias of the valve characteristics to the open position increased with the inter-leaflet gap in the resting position and with valve inflation pressure. The characteristics of the simulated valve were compared with those specified in an existing lumped-parameter model of one or more collecting lymphangions and used to estimate a revised value for the constant in that model which controls the rate of valve opening/closure with variation in applied pressure difference. The effects of the revised value on the lymph pumping efficacy predicted by the lumped-parameter model were evaluated.

Lymphatic remodelling in response to lymphatic injury in the hind limbs of sheep

Contractile activity in the lymphatic vasculature is essential for maintaining fluid balance within organs and tissues. However, the mechanisms by which collecting lymphatics adapt to changes in fluid load and how these adaptations influence lymphatic contractile activity are unknown. Here we report a model of lymphatic injury based on the ligation of one of two parallel lymphatic vessels in the hind limb of sheep and the evaluation of structural and functional changes in the intact, remodelling lymphatic vessel over a 42-day period. We show that the remodelled lymphatic vessel displayed increasing intrinsic contractile frequency, force generation and vessel compliance, as well as decreasing flow-mediated contractile inhibition via the enzyme endothelial nitric oxide synthase. A computational model of a chain of lymphatic contractile segments incorporating these adaptations predicted increases in the flow-generation capacity of the remodelled vessel at the expense of normal mitochondrial function and elevated oxidative stress within the lymphatic muscle. Our findings may inform interventions for mitigating lymphatic muscle fatigue in patients with dysfunctional lymphatics.

A computational model of a network of initial lymphatics and pre-collectors with permeable interstitium

Initial lymphatic vessels are made up of overlapped endothelial cells that act as unidirectional valves enabling one-way drainage of tissue fluid into the lumen of the initial lymphatics when there is a favourable pressure gradient. Initial lymphatics subsequently drain this fluid into the collecting lymphatics. This paper describes a computational model for a network of passive rat mesenteric lymphatic vessels with sparse secondary valves. The network was simulated with the secondary valves both operational and non-operational. The effects on the cycle-mean outflow-rate from the network of both inflammation and the resistance of the surrounding interstitium were considered. The cycle-mean outflow-rate is sensitive to vessel stiffness. If the influence of primary-valve resistance is reduced relative to that of interstitial resistance and intravascular resistance, there is no absolute advantage of extrinsic pumping, since maximum outflow-rate occurs when vessels are rigid. However, there is relative advantage, in that the outflow-rate at intermediate stiffness is higher with the secondary valves functioning than when they are deactivated. If primary-valve resistance dominates, then extrinsic pumping of non-rigid vessels provides absolute advantage. The nonlinear relation between pressure drop and flow-rate of the endothelial primary valves, combined with downstream compliance and pulsatile external pressure, constitutes a separate mechanism of pumping. By enabling the consideration of interactions between multiple phenomena (primary valves, secondary valves, a real network geometry with multiple branches, deformable vessel walls, interstitial resistance and external pressures), the model offers a perspective for delineating physiological phenomena that have not yet been fully linked to the biomechanics of fluid flow through initial lymphatic networks.

The effects of valve leaflet mechanics on lymphatic pumping assessed using numerical simulations

The lymphatic system contains intraluminal leaflet valves that function to bias lymph flow back towards the heart. These valves are present in the collecting lymphatic vessels, which generally have lymphatic muscle cells and can spontaneously pump fluid. Recent studies have shown that the valves are open at rest, can allow some backflow, and are a source of nitric oxide (NO). To investigate how these valves function as a mechanical valve and source of vasoactive species to optimize throughput, we developed a mathematical model that explicitly includes Ca²⁺ -modulated contractions, NO production and valve structures. The 2D lattice Boltzmann model includes an initial lymphatic vessel and a collecting lymphangion embedded in a porous tissue. The lymphangion segment has mechanically-active vessel walls and is flanked by deformable valves. Vessel wall motion is passively affected by fluid pressure, while active contractions are driven by intracellular Ca²⁺ fluxes. The model reproduces NO and Ca²⁺ dynamics, valve motion and fluid drainage from tissue. We find that valve structural properties have dramatic effects on performance, and that valves with a stiffer base and flexible tips produce more stable cycling. In agreement with experimental observations, the valves are a major source of NO. Once initiated, the contractions are spontaneous and self-sustained, and the system exhibits interesting non-linear dynamics. For example, increased fluid pressure in the tissue or decreased lymph pressure at the outlet of the system produces high shear stress and high levels of NO, which inhibits contractions. On the other hand, a high outlet pressure opposes the flow, increasing the luminal pressure and the radius of the vessel, which results in strong contractions in response to mechanical stretch of the wall. We also find that the location of contraction initiation is affected by the extent of backflow through the valves.

Lymphatic Vessel Network Structure and Physiology

The lymphatic system is comprised of a network of vessels interrelated with lymphoid tissue, which has the holistic function to maintain the local physiologic environment for every cell in all tissues of the body. The lymphatic system maintains extracellular fluid homeostasis favorable for optimal tissue function, removing substances that arise due to metabolism or cell death, and optimizing immunity against bacteria, viruses, parasites, and other antigens. This article provides a comprehensive review of important findings over the past century along with recent advances in the understanding of the anatomy and physiology of lymphatic vessels, including tissue/organ specificity, development, mechanisms of lymph formation and transport, lymphangiogenesis, and the roles of lymphatics in disease. © 2019 American Physiological Society. Compr Physiol 9:207-299, 2019.

Effect of Axial Stretch on Lumen Collapse of Arteries

The stability of the arteries under in vivo pressure and axial tension loads is essential to normal arterial function, and lumen collapse due to buckling can hinder the blood flow. The objective of this study was to develop the lumen buckling equation for nonlinear anisotropic thick-walled arteries to determine the effect of axial tension. The theoretical equation was developed using exponential Fung strain function, and the effects of axial tension and residual stress on the critical buckling pressure were illustrated for porcine coronary arteries. The buckling behavior was also simulated using finite-element analysis. Our results demonstrated that lumen collapse of arteries could occur when the transmural pressure is negative and exceeded a critical value. This value depends upon the axial stretch ratio and material properties of the arterial wall. Axial tensions show a biphasic effect on the critical buckling pressure. The lumen aspect ratio of arteries increases nonlinearly with increasing external pressure beyond the critical value as the lumen collapses. These results enhance our understanding of artery lumen collapse behavior.

A one-dimensional mathematical model of collecting lymphatics coupled with an electro-fluid-mechanical contraction model and valve dynamics

We propose a one-dimensional model for collecting lymphatics coupled with a novel Electro-Fluid-Mechanical Contraction (EFMC) model for dynamical contractions, based on a modified FitzHugh-Nagumo model for action potentials. The one-dimensional model for a deformable lymphatic vessel is a nonlinear system of hyperbolic Partial Differential Equations (PDEs). The EFMC model combines the electrical activity of lymphangions (action potentials) with fluid-mechanical feedback (circumferential stretch of the lymphatic wall and wall shear stress) and lymphatic vessel wall contractions. The EFMC model is governed by four Ordinary Differential Equations (ODEs) and phenomenologically relies on: (1) environmental calcium influx, (2) stretch-activated calcium influx, and (3) contraction inhibitions induced by wall shear stresses. We carried out a stability analysis of the stationary state of the EFMC model. Contractions turn out to be triggered by the instability of the stationary state. Overall, the EFMC model allows emulating the influence of pressure and wall shear stress on the frequency of contractions observed experimentally. Lymphatic valves are modelled by extending an existing lumped-parameter model for blood vessels. Modern numerical methods are employed for the one-dimensional model (PDEs), for the EFMC model and valve dynamics (ODEs). Adopting the geometrical structure of collecting lymphatics from rat mesentery, we apply the full mathematical model to a carefully selected suite of test problems inspired by experiments. We analysed several indices of a single lymphangion for a wide range of upstream and downstream pressure combinations which included both favourable and adverse pressure gradients. The most influential model parameters were identified by performing two sensitivity analyses for favourable and adverse pressure gradients.

Demonstration and Analysis of the Suction Effect for Pumping Lymph from Tissue Beds at Subatmospheric Pressure

Many tissues exhibit subatmospheric interstitial pressures under normal physiologic conditions. The mechanisms by which the lymphatic system extracts fluid from these tissues against the overall pressure gradient are unknown. We address this important physiologic issue by combining experimental measurements of contractile function and pressure generation with a previously validated mathematical model. We provide definitive evidence for the existence of 'suction pressure' in collecting lymphatic vessels, which manifests as a transient drop in pressure downstream of the inlet valve following contraction. This suction opens the inlet valve and is required for filling in the presence of low upstream pressure. Positive transmural pressure is required for this suction, providing the energy required to reopen the vessel. Alternatively, external vessel tethering can serve the same purpose when the transmural pressure is negative. Suction is transmitted upstream, allowing fluid to be drawn in through initial lymphatics. Because suction plays a major role in fluid entry to the lymphatics and is affected by interstitial pressure, our results introduce the phenomenon as another important factor to consider in the study of lymphoedema and its treatment.

The passive biomechanics of human pelvic collecting lymphatic vessels

The lymphatic system has a major significance in the metastatic pathways in women's cancers. Lymphatic pumping depends on both extrinsic and intrinsic mechanisms, and the mechanical behavior of lymphatic vessels regulates the function of the system. However, data on the mechanical properties and function of human lymphatics are lacking. Our aim is to characterize, for the first time, the passive biomechanical behavior of human collecting lymphatic vessels removed at pelvic lymph node dissection during primary debulking surgeries for epithelial ovarian cancer. Isolated vessels were cannulated and then pressurized at varying levels of applied axial stretch in a calcium-free Krebs buffer. Pressurized vessels were then imaged using multi-photon microscopy for collagen-elastin structural composition and fiber orientation. Both pressure-diameter and force-elongation responses were highly nonlinear, and axial stretching of the vessel served to decrease diameter at constant pressure. Pressure-diameter behavior for the human vessels is very similar to data from rat mesenteric vessels, though the human vessels were approximately 10× larger than those from rats. Multiphoton microscopy revealed the vessels to be composed of an inner layer of elastin with an outer layer of aligned collagen fibers. This is the first study that successfully described the passive biomechanical response and composition of human lymphatic vessels in patients with ovarian cancer. Future work should expand on this knowledge base with investigations of vessels from other anatomical locations, contractile behavior, and the implications on metastatic cell transport.

Opportunities for Studying the Hydrodynamic Context for Breast Cancer Cell Spread Through Lymph Flow

The lymphatic system serves as the primary route for the metastatic spread of breast cancer cells (BCCs). A scarcity of information exists with regard to the advection of BCCs in lymph flow and a fundamental understanding of the response of BCCs to the forces in the lymphatics needs to be established. This review summarizes the flow environment metastatic BCCs are exposed to in the lymphatics. Special attention is paid to the behavior of cells/particles in microflows in an attempt to elucidate the behavior of BCCs under lymph flow conditions (Reynolds number <1).

Valve-related modes of pump failure in collecting lymphatics: numerical and experimental investigation

Lymph is transported along collecting lymphatic vessels by intrinsic and extrinsic pumping. The walls have muscle of a type intermediate between blood-vascular smooth muscle and myocardium; a contracting segment between two valves (a lymphangion) constitutes a pump. This intrinsic mechanism is investigated ex vivo in isolated, spontaneously contracting, perfused segments subjected to controlled external pressures. The reaction to varying afterload is probed by slowly ramping up the outlet pressure until pumping fails. Often the failure occurs when the contraction raises intra-lymphangion pressure insufficiently to overcome the outlet pressure, open the outlet valve and cause ejection, but many segments fail by other means, the mechanisms of which are not clear. We here elucidate those mechanisms by resort to a numerical model. Experimental observations are paired with comparable findings from computer simulations, using a lumped-parameter model that incorporates previously measured valve properties, plus new measurements of active contractile and passive elastic properties, and the dependence of contraction frequency on transmural pressure, all taken from isobaric twitch contraction experiments in the same vessel. Surprisingly, the model predicts seven different possible modes of pump failure, each defined by a different sequence of valve events, with their occurrence depending on the parameter values and boundary conditions. Some, but not all, modes were found experimentally. Further model investigation reveals routes by which a vessel exhibiting one mode of failure might under altered circumstances exhibit another.

A lumped parameter model of mechanically mediated acute and long-term adaptations of contractility and geometry in lymphatics for characterization of lymphedema

A primary purpose of the lymphatic system is to transport fluid from peripheral tissues to the central venous system in order to maintain tissue-fluid balance. Failure to perform this task results in lymphedema marked by swelling of the affected limb as well as geometric remodeling and reduced contractility of the affected lymphatic vessels. The mechanical environment has been implicated in the regulation of lymphatic contractility, but it is unknown how changes in the mechanical environment are related to loss of contractile function and remodeling of the tissue. The purpose of this paper was to introduce a new theoretical framework for acute and long-term adaptations of lymphatic vessels to changes in mechanical loading. This theoretical framework combines a simplified version of a published lumped parameter model for lymphangion function and lymph transport, a published microstructurally motivated constitutive model for the active and passive mechanical behavior of isolated rat thoracic ducts, and novel models for acute mechanically mediated vasoreactive adaptations and long-term volumetric growth to simulate changes in muscle contractility and geometry of a single isolated rat thoracic duct in response to a sustained elevation in afterload. The illustrative examples highlight the potential role of the mechanical environment in the acute maintenance of contractility and long-term geometric remodeling, presumably aimed at meeting fluid flow demands while also maintaining mechanical homeostasis. Results demonstrate that contractility may adapt in response to shear stress to meet fluid flow demands and show that pressure-induced long-term geometric remodeling may attenuate these adaptations and reduce fluid flow. The modeling framework and illustrative simulations help suggest relevant experiments that are necessary to accurately quantify and predict the acute and long-term adaptations of lymphangions to altered mechanical loading.

Endothelial Cell Responses to Biomechanical Forces in Lymphatic Vessels

SIGNIFICANCE

Lymphatic vessels are important components of the cardiovascular and immune systems. They contribute both to the maintenance of normal homeostasis and to many pathological conditions, such as cancer and inflammation. The lymphatic vasculature is subjected to a variety of biomechanical forces, including fluid shear stress and vessel circumferential stretch.

RECENT ADVANCES

This review will discuss recent advances in our understanding of biomechanical forces in lymphatic vessels and their role in mammalian lymphatic vascular development and function.

CRITICAL ISSUES

We will highlight the importance of fluid shear stress generated by lymph flow in organizing the lymphatic vascular network. We will also describe how mutations in mechanosensitive genes lead to lymphatic vascular dysfunction.

FUTURE DIRECTIONS

Better understanding of how biomechanical and biochemical stimuli are perceived and interpreted by lymphatic endothelial cells is important for targeting regulation of lymphatic function in health and disease. Important remaining critical issues and future directions in the field will be discussed in this review. Antioxid. Redox Signal. 25, 451-465.

Estimation of the Pressure Drop Required for Lymph Flow through Initial Lymphatic Networks

BACKGROUND

Lymphatic function is critical for maintaining interstitial fluid balance and is linked to multiple pathological conditions. While smooth muscle contractile mechanisms responsible for fluid flow through collecting lymphatic vessels are well studied, how fluid flows into and through initial lymphatic networks remains poorly understood. The objective of this study was to estimate the pressure difference needed for flow through an intact initial lymphatic network.

METHODS AND RESULTS

Pressure drops were computed for real and theoretical networks with varying branch orders using a segmental Poiseuille flow model. Vessel geometries per branch order were based on measurements from adult Wistar rat mesenteric initial lymphatic networks. For computational predications based on real network geometries and combinations of low or high output velocities (2 mm/s, 4 mm/s) and viscosities (1 cp, 1.5 cp), pressure drops were estimated to range 0.31-2.57 mmHg. The anatomical data for the real networks were also used to create a set of theoretical networks in order to identify possible minimum and maximum pressure drops. The pressure difference range for the theoretical networks was 0.16-3.16 mmHg.

CONCLUSIONS

The results support the possibility for suction pressures generated from cyclic smooth muscle contractions of upstream collecting lymphatics being sufficient for fluid flow through an initial lymphatic network.

Pump function curve shape for a model lymphatic vessel

The transport capacity of a contractile segment of lymphatic vessel is defined by its pump function curve relating mean flow-rate and adverse pressure difference. Numerous system characteristics affect curve shape and the magnitude of the generated flow-rates and pressures. Some cannot be varied experimentally, but their separate and interacting effects can be systematically revealed numerically. This paper explores variations in the rate of change of active tension and the form of the relation between active tension and muscle length, factors not known from experiment to functional precision. Whether the pump function curve bends toward or away from the origin depends partly on the curvature of the passive pressure-diameter relation near zero transmural pressure, but rather more on the form of the relation between active tension and muscle length. A pump function curve bending away from the origin defines a well-performing pump by maximum steady output power. This behaviour is favoured by a length/active-tension relationship which sustains tension at smaller lengths. Such a relationship also favours high peak mechanical efficiency, defined as output power divided by the input power obtained from the lymphangion diameter changes and active-tension time-course. The results highlight the need to pin down experimentally the form of the length/active-tension relationship.

Network Scale Modeling of Lymph Transport and Its Effective Pumping Parameters

The lymphatic system is an open-ended network of vessels that run in parallel to the blood circulation system. These vessels are present in almost all of the tissues of the body to remove excess fluid. Similar to blood vessels, lymphatic vessels are found in branched arrangements. Due to the complexity of experiments on lymphatic networks and the difficulty to control the important functional parameters in these setups, computational modeling becomes an effective and essential means of understanding lymphatic network pumping dynamics. Here we aimed to determine the effect of pumping coordination in branched network structures on the regulation of lymph flow. Lymphatic vessel networks were created by building upon our previous lumped-parameter model of lymphangions in series. In our network model, each vessel is itself divided into multiple lymphangions by lymphatic valves that help maintain forward flow. Vessel junctions are modeled by equating the pressures and balancing mass flows. Our results demonstrated that a 1.5 s rest-period between contractions optimizes the flow rate. A time delay between contractions of lymphangions at the junction of branches provided an advantage over synchronous pumping, but additional time delays within individual vessels only increased the flow rate for adverse pressure differences greater than 10.5 cmH2O. Additionally, we quantified the pumping capability of the system under increasing levels of steady transmural pressure and outflow pressure for different network sizes. We observed that peak flow rates normally occurred under transmural pressures between 2 to 4 cmH2O (for multiple pressure differences and network sizes). Networks with 10 lymphangions per vessel had the highest pumping capability under a wide range of adverse pressure differences. For favorable pressure differences, pumping was more efficient with fewer lymphangions. These findings are valuable for translating experimental measurements from the single lymphangion level to tissue and organ scales.

Consequences of intravascular lymphatic valve properties: a study of contraction timing in a multi-lymphangion model

The observed properties of valves in collecting lymphatic vessels include transmural pressure-dependent bias to the open state and hysteresis. The bias may reduce resistance to flow when the vessel is functioning as a conduit. However, lymphatic pumping implies a streamwise increase in mean pressure across each valve, suggesting that the bias is then potentially unhelpful. Lymph pumping by a model of several collecting lymphatic vessel segments (lymphangions) in series, which incorporated these properties, was investigated under conditions of adverse pressure difference while varying the refractory period between active muscular contractions and the inter-lymphangion contraction delay. It was found that many combinations of the timing parameters and the adverse pressure difference led to one or more intermediate valves remaining open instead of switching between open and closed states during repetitive contraction cycles. Cyclic valve switching was reliably indicated if the mean pressure in a lymphangion over a cycle was higher than that in the lymphangion upstream, but either lack of or very brief valve closure could cause mean pressure to be lower downstream. Widely separated combinations of refractory period and delay time were found to produce the greatest flow-rate for a given pressure difference. The efficiency of pumping was always maximized by a long refractory period and lymphangion contraction starting when the contraction of the lymphangion immediately upstream was peaking. By means of an ex vivo experiment, it was verified that intermediate valves in a chain of pumping lymphangions can remain open, while the lymphangions on either side of the open valve continue to execute contractions.

Effects of dynamic shear and transmural pressure on wall shear stress sensitivity in collecting lymphatic vessels

Given the known mechanosensitivity of the lymphatic vasculature, we sought to investigate the effects of dynamic wall shear stress (WSS) on collecting lymphatic vessels while controlling for transmural pressure. Using a previously developed ex vivo lymphatic perfusion system (ELPS) capable of independently controlling both transaxial pressure gradient and average transmural pressure on an isolated lymphatic vessel, we imposed a multitude of flow conditions on rat thoracic ducts, while controlling for transmural pressure and measuring diameter changes. By gradually increasing the imposed flow through a vessel, we determined the WSS at which the vessel first shows sign of contraction inhibition, defining this point as the shear stress sensitivity of the vessel. The shear stress threshold that triggered a contractile response was significantly greater at a transmural pressure of 5 cmH2O (0.97 dyne/cm(2)) than at 3 cmH2O (0.64 dyne/cm(2)). While contraction frequency was reduced when a steady WSS was applied, this inhibition was reversed when the applied WSS oscillated, even though the mean wall shear stresses between the conditions were not significantly different. When the applied oscillatory WSS was large enough, flow itself synchronized the lymphatic contractions to the exact frequency of the applied waveform. Both transmural pressure and the rate of change of WSS have significant impacts on the contractile response of lymphatic vessels to flow. Specifically, time-varying shear stress can alter the inhibition of phasic contraction frequency and even coordinate contractions, providing evidence that dynamic shear could play an important role in the contractile function of collecting lymphatic vessels.

Mechanobiological oscillators control lymph flow

The ability of cells to sense and respond to physical forces has been recognized for decades, but researchers are only beginning to appreciate the fundamental importance of mechanical signals in biology. At the larger scale, there has been increased interest in the collective organization of cells and their ability to produce complex, "emergent" behaviors. Often, these complex behaviors result in tissue-level control mechanisms that manifest as biological oscillators, such as observed in fireflies, heartbeats, and circadian rhythms. In many cases, these complex, collective behaviors are controlled--at least in part--by physical forces imposed on the tissue or created by the cells. Here, we use mathematical simulations to show that two complementary mechanobiological oscillators are sufficient to control fluid transport in the lymphatic system: Ca(2+)-mediated contractions can be triggered by vessel stretch, whereas nitric oxide produced in response to the resulting fluid shear stress causes the lymphatic vessel to relax locally. Our model predicts that the Ca(2+) and NO levels alternate spatiotemporally, establishing complementary feedback loops, and that the resulting phasic contractions drive lymph flow. We show that this mechanism is self-regulating and robust over a range of fluid pressure environments, allowing the lymphatic vessels to provide pumping when needed but remain open when flow can be driven by tissue pressure or gravity. Our simulations accurately reproduce the responses to pressure challenges and signaling pathway manipulations observed experimentally, providing an integrated conceptual framework for lymphatic function.

Mechanobiology of lymphatic contractions

The lymphatic system is responsible for controlling tissue fluid pressure by facilitating flow of lymph (i.e. the plasma and cells that enter the lymphatic system). Because lymph contains cells of the immune system, its transport is not only important for fluid homeostasis, but also immune function. Lymph drainage can occur via passive flow or active pumping, and much research has identified the key biochemical and mechanical factors that affect output. Although many studies and reviews have addressed how tissue properties and fluid mechanics (i.e. pressure gradients) affect lymph transport [1-3] there is less known about lymphatic mechanobiology. As opposed to passive mechanical properties, mechanobiology describes the active coupling of mechanical signals and biochemical pathways. Lymphatic vasomotion is the result of a fascinating system affected by mechanical forces exerted by the flowing lymph, including pressure-induced vessel stretch and flow-induced shear stresses. These forces can trigger or modulate biochemical pathways important for controlling the lymphatic contractions. Here, I review the current understanding of lymphatic vessel function, focusing on vessel mechanobiology, and summarize the prospects for a comprehensive understanding that integrates the mechanical and biomechanical control mechanisms in the lymphatic system.

Mechanical forces and lymphatic transport

This review examines the current understanding of how the lymphatic vessel network can optimize lymph flow in response to various mechanical forces. Lymphatics are organized as a vascular tree, with blind-ended initial lymphatics, precollectors, prenodal collecting lymphatics, lymph nodes, postnodal collecting lymphatics and the larger trunks (thoracic duct and right lymph duct) that connect to the subclavian veins. The formation of lymph from interstitial fluid depends heavily on oscillating pressure gradients to drive fluid into initial lymphatics. Collecting lymphatics are segmented vessels with unidirectional valves, with each segment, called a lymphangion, possessing an intrinsic pumping mechanism. The lymphangions propel lymph forward against a hydrostatic pressure gradient. Fluid is returned to the central circulation both at lymph nodes and via the larger lymphatic trunks. Several recent developments are discussed, including evidence for the active role of endothelial cells in lymph formation; recent developments on how inflow pressure, outflow pressure, and shear stress affect the pump function of the lymphangion; lymphatic valve gating mechanisms; collecting lymphatic permeability; and current interpretations of the molecular mechanisms within lymphatic endothelial cells and smooth muscle. An improved understanding of the physiological mechanisms by which lymphatic vessels sense mechanical stimuli, integrate the information, and generate the appropriate response is key for determining the pathogenesis of lymphatic insufficiency and developing treatments for lymphedema.

Confocal image-based computational modeling of nitric oxide transport in a rat mesenteric lymphatic vessel

The lymphatic system plays important roles in protein and solute transport as well as in the immune system. Its functionality is vital to proper homeostasis and fluid balance. Lymph may be propelled by intrinsic (active) vessel pumping or passive compression from external tissue movement. With regard to the former, nitric oxide (NO) is known to play an important role modulating lymphatic vessel contraction and vasodilation. Lymphatic endothelial cells (LECs) are sensitive to shear, and increases in flow have been shown to cause enhanced production of NO by LECs. Additionally, high concentrations of NO have been experimentally observed in the sinus region of mesenteric lymphatic vessels. A computational flow and mass transfer model using physiologic geometries obtained from confocal images of a rat mesenteric lymphatic vessel was developed to determine the characteristics of NO transport in the lymphatic flow regime. Both steady and unsteady analyses were performed. Production of NO was shear-dependent; basal cases using constant production were also generated. Simulations revealed areas of flow stagnation adjacent to the valve leaflets, suggesting the high concentrations observed here experimentally are due to minimal convection in this region. LEC sensitivity to shear was found to alter the concentration of NO in the vessel, and the convective forces were found to profoundly affect the concentration of NO at a Péclet value greater than approximately 61. The quasisteady analysis was able to resolve wall shear stress within 0.15% of the unsteady case. However, the percent difference between unsteady and quasisteady conditions was higher for NO concentration (6.7%). We have shown high NO concentrations adjacent to the valve leaflets are most likely due to flow-mediated processes rather than differential production by shear-sensitive LECs. Additionally, this model supports experimental findings of shear-dependent production, since removing shear dependence resulted in concentrations that are physiologically counterintuitive. Understanding the transport mechanisms and flow regimes in the lymphatic vasculature could help in the development of therapeutics to treat lymphatic disorders.

Ex vivo lymphatic perfusion system for independently controlling pressure gradient and transmural pressure in isolated vessels

In addition to external forces, collecting lymphatic vessels intrinsically contract to transport lymph from the extremities to the venous circulation. As a result, the lymphatic endothelium is routinely exposed to a wide range of dynamic mechanical forces, primarily fluid shear stress and circumferential stress, which have both been shown to affect lymphatic pumping activity. Although various ex vivo perfusion systems exist to study this innate pumping activity in response to mechanical stimuli, none are capable of independently controlling the two primary mechanical forces affecting lymphatic contractility: transaxial pressure gradient, [Formula: see text], which governs fluid shear stress; and average transmural pressure, [Formula: see text], which governs circumferential stress. Hence, the authors describe a novel ex vivo lymphatic perfusion system (ELPS) capable of independently controlling these two outputs using a linear, explicit model predictive control (MPC) algorithm. The ELPS is capable of reproducing arbitrary waveforms within the frequency range observed in the lymphatics in vivo, including a time-varying [Formula: see text] with a constant [Formula: see text], time-varying [Formula: see text] and [Formula: see text], and a constant [Formula: see text] with a time-varying [Formula: see text]. In addition, due to its implementation of syringes to actuate the working fluid, a post-hoc method of estimating both the flow rate through the vessel and fluid wall shear stress over multiple, long (5 s) time windows is also described.

Parameter sensitivity analysis of a lumped-parameter model of a chain of lymphangions in series

Any disruption of the lymphatic system due to trauma or injury can lead to edema. There is no effective cure for lymphedema, partly because predictive knowledge of lymphatic system reactions to interventions is lacking. A well-developed model of the system could greatly improve our understanding of its function. Lymphangions, defined as the vessel segment between two valves, are the individual pumping units. Based on our previous lumped-parameter model of a chain of lymphangions, this study aimed to identify the parameters that affect the system output the most using a sensitivity analysis. The system was highly sensitive to minimum valve resistance, such that variations in this parameter caused an order-of-magnitude change in time-average flow rate for certain values of imposed pressure difference. Average flow rate doubled when contraction frequency was increased within its physiological range. Optimum lymphangion length was found to be some 13-14.5 diameters. A peak of time-average flow rate occurred when transmural pressure was such that the pressure-diameter loop for active contractions was centered near maximum passive vessel compliance. Increasing the number of lymphangions in the chain improved the pumping in the presence of larger adverse pressure differences. For a given pressure difference, the optimal number of lymphangions increased with the total vessel length. These results indicate that further experiments to estimate valve resistance more accurately are necessary. The existence of an optimal value of transmural pressure may provide additional guidelines for increasing pumping in areas affected by edema.

Development of a model of a multi-lymphangion lymphatic vessel incorporating realistic and measured parameter values

Our published model of a lymphatic vessel consisting of multiple actively contracting segments between non-return valves has been further developed by the incorporation of properties derived from observations and measurements of rat mesenteric vessels. These included (1) a refractory period between contractions, (2) a highly nonlinear form for the passive part of the pressure-diameter relationship, (3) hysteretic and transmural-pressure-dependent valve opening and closing pressure thresholds and (4) dependence of active tension on muscle length as reflected in local diameter. Experimentally, lymphatic valves are known to be biased to stay open. In consequence, in the improved model, vessel pumping of fluid suffers losses by regurgitation, and valve closure is dependent on backflow first causing an adverse valve pressure drop sufficient to reach the closure threshold. The assumed resistance of an open valve therefore becomes a critical parameter, and experiments to measure this quantity are reported here. However, incorporating this parameter value, along with other parameter values based on existing measurements, led to ineffective pumping. It is argued that the published measurements of valve-closing pressure threshold overestimate this quantity owing to neglect of micro-pipette resistance. An estimate is made of the extent of the possible resulting error. Correcting by this amount, the pumping performance is improved, but still very inefficient unless the open-valve resistance is also increased beyond the measured level. Arguments are given as to why this is justified, and other areas where experimental data are lacking are identified. The model is capable of future adaptation as new experimental data appear.

Incorporating measured valve properties into a numerical model of a lymphatic vessel

An existing lumped-parameter model of multiple lymphangions (lymphatic vascular segments) in series is adapted for the incorporation of recent physiological measurements of lymphatic vascular properties. The new data show very marked nonlinearity of the passive pressure-diameter relation during distension, relative to comparable blood vessels, and complex valve behaviour. Since lymph is transported as a result of either the active contraction or the passive squeezing of vascular segments situated between two one-way valves, the performance of these valves is of primary importance. The valves display hysteresis (the opening and closing pressure drop thresholds differ), a bias to staying open (both state changes occur when the trans-valve pressure drop is adverse) and pressure-drop threshold dependence on transmural pressure. These properties, in combination with the strong nonlinearity that valve operation represents, have in turn caused intriguing numerical problems in the model, and we describe numerical stratagems by which we have overcome the problems. The principal problem is also generalised into a relatively simple mathematical example, for which solution detail is provided using two different solvers.

Modelling the lymphatic system: challenges and opportunities

The lymphatic system is a vital part of the circulatory and immune systems, and plays an important role in homeostasis by controlling extracellular fluid volume and in combating infection. Nevertheless, there is a notable disparity in terms of research effort expended in relation to the treatment of lymphatic diseases in contrast to the cardiovascular system. While similarities to the cardiovascular system exist, there are considerable differences in their anatomy and physiology. This review outlines some of the challenges and opportunities for those engaged in modelling biological systems. The study of the lymphatic system is still in its infancy, the vast majority of the models presented in the literature to date having been developed since 2003. The number of distinct models and their variants are few in number, and only one effort has been made thus far to study the entire lymphatic network; elements of the lymphatic system such as the nodes, which act as pumps and reservoirs, have not been addressed by mathematical models. Clearly, more work will be necessary in combination with experimental verification in order to progress and update the knowledge on the function of the lymphatic system. As our knowledge and understanding of its function increase, new and more effective treatments of lymphatic diseases are bound to emerge.

Engineering the Lymphatic System

The recent advances in our understanding of lymphatic physiology and the role of the lymphatics in actively regulating fluid balance, lipid transport, and immune cell trafficking has been furthered in part through innovations in imaging, tissue engineering, quantitative biology, biomechanics, and computational modeling. Interdisciplinary and bioengineering approaches will continue to be crucial to the progression of the field, given that lymphatic biology and function are intimately woven with the local microenvironment and mechanical loads experienced by the vessel. This is particularly the case in lymphatic diseases such as lymphedema where the microenvironment can be drastically altered by tissue fibrosis and adipocyte accumulation. In this review we will highlight contributions engineering and mechanics have made to lymphatic physiology and will discuss areas that will be important for future research.

Determinants of valve gating in collecting lymphatic vessels from rat mesentery

Secondary lymphatic valves are essential for minimizing backflow of lymph and are presumed to gate passively according to the instantaneous trans-valve pressure gradient. We hypothesized that valve gating is also modulated by vessel distention, which could alter leaflet stiffness and coaptation. To test this hypothesis, we devised protocols to measure the small pressure gradients required to open or close lymphatic valves and determine if the gradients varied as a function of vessel diameter. Lymphatic vessels were isolated from rat mesentery, cannulated, and pressurized using a servo-control system. Detection of valve leaflet position simultaneously with diameter and intraluminal pressure changes in two-valve segments revealed the detailed temporal relationships between these parameters during the lymphatic contraction cycle. The timing of valve movements was similar to that of cardiac valves, but only when lymphatic vessel afterload was elevated. The pressure gradients required to open or close a valve were determined in one-valve segments during slow, ramp-wise pressure elevation, either from the input or output side of the valve. Tests were conducted over a wide range of baseline pressures (and thus diameters) in passive vessels as well as in vessels with two levels of imposed tone. Surprisingly, the pressure gradient required for valve closure varied >20-fold (0.1-2.2 cmH(2)O) as a passive vessel progressively distended. Similarly, the pressure gradient required for valve opening varied sixfold with vessel distention. Finally, our functional evidence supports the concept that lymphatic muscle tone exerts an indirect effect on valve gating.

A model of a radially expanding and contracting lymphangion

The lymphatic system is an extensive vascular network featuring valves and contractile walls that pump interstitial fluid and plasma proteins back to the main circulation. Immune function also relies on the lymphatic system's ability to transport white blood cells. Failure to drain and pump this excess fluid results in edema characterized by fluid retention and swelling of limbs. It is, therefore, important to understand the mechanisms of fluid transport and pumping of lymphatic vessels. Unfortunately, there are very few studies in this area, most of which assume Poiseuille flow conditions. In vivo observations reveal that these vessels contract strongly, with diameter changes of the order of magnitude of the diameter itself over a cycle that lasts typically 2-3s. The radial velocity of the contracting vessel is on the order of the axial fluid velocity, suggesting that modeling flow in these vessels with a Poiseuille model is inappropriate. In this paper, we describe a model of a radially expanding and contracting lymphatic vessel and investigate the validity of assuming Poiseuille flow to estimate wall shear stress, which is presumably important for lymphatic endothelial cell mechanotransduction. Three different wall motions, periodic sinusoidal, skewed sinusoidal and physiologic wall motions, were investigated with steady and unsteady parabolic inlet velocities. Despite high radial velocities resulting from the wall motion, wall shear stress values were within 4% of quasi-static Poiseuille values. Therefore, Poiseuille flow is valid for the estimation of wall shear stress for the majority of the lymphangion contractile cycle.